Sobolev orthogonal polynomials:
the discrete-continuous case
Manuel Alfaro∗ , Teresa E. Pérez† ,
Miguel A. Piñar† and M. Luisa Rezola∗
Abstract
In this paper, we study orthogonal polynomials with respect to the
bilinear form
(N )
BS (f, g) = F (c)AG(c)T + hu, f (N ) g (N ) i,
where u is a quasi-definite (or regular) linear functional on the linear
space P of real polynomials, c is a real number, N is a positive integer
number, A is a symmetric N ×N real matrix such that each of its principal submatrices are regular, and F (c) = (f (c), f 0 (c), . . . , f (N −1) (c)),
G(c) = (g(c), g 0 (c), . . . , g (N −1) (c)). For these non–standard orthogonal
polynomials, algebraic and differential properties are obtained, as well
as their representation in terms of the standard orthogonal polynomials
associated with u.
Running title: Discrete–continuous Sobolev polynomials,
1
1991 Mathematics Subject Classification: 33C45, 42C05
Key words and phrases: Sobolev orthogonal polynomials, Classical orthogonal
polynomials.
2
1
1
Introduction
It is well known (see [12]) that the monic generalized Laguerre polynomials
(α)
{Ln }n satisfy, for any real value of α, the three-term recurrence relation
(α)
(α)
(α) (α)
(α)
xL(α)
n (x) = Ln+1 (x) + βn Ln (x) + γn Ln−1 (x),
(α)
(α)
L−1 (x) = 0,
L0 (x) = 1,
where
βn(α) = 2n + α + 1,
γn(α) = n(n + α).
(α)
Whenever α is not a negative integer number, we have γn 6= 0 for all n ≥
(α)
1 and Favard’s theorem (see [2], p. 21) ensures that the sequence {Ln }n
is orthogonal with respect to a quasi–definite linear functional. Besides, if
α > −1 the functional is definite positive and the polynomials are orthogonal
with respect to the weight xα e−x on the interval (0, +∞). For α a negative
(α)
integer number, since γn vanishes for some value of n, no orthogonality
results can be deduced from Favard’s theorem.
In the last few years, orthogonal polynomials with respect to an inner
product involving derivatives (the so–called Sobolev orthogonal polynomials) have been the object of increasing interest and in this context, the case
(α)
{Ln }n with α a negative integer number has been solved. More precisely,
Kwon and Littlejohn, in [5], established the orthogonality of the generalized
(−k)
Laguerre polynomials {Ln }n , k ≥ 1, with respect to a Sobolev inner
product of the form
hf, gi = F (0)AG(0)T +
Z
+∞
f (k) (x)g (k) (x)e−x dx
0
with A a symmetric k × k real matrix, F (0) = (f (0), f 0 (0), . . . , f (k−1) (0)),
and G(0) = (g(0), g 0 (0), . . . , g (k−1) (0)). The particular case k = 1 had been
considered by the same authors in a previous paper, (see [6]).
In [10], Pérez and Piñar gave an unified approach to the orthogonality of
the generalized Laguerre polynomials, for any real value of the parameter α
by proving their orthogonality with respect to a Sobolev non–diagonal inner
product. So, they obtained the following result:
(N,α)
Theorem ([10]) Let (., .)S
(N,α)
(f, g)S
be the Sobolev inner product defined by
Z
=
+∞
F (x)AG(x)T xα e−x dx,
0
2
where the (i, j)-entry of A is given by
min{i,j}
mi,j (N ) =
X
p=0
(−1)i+j
N −p
i−p
!
!
N −p
,
j−p
0 ≤ i, j ≤ N,
F (x) = (f (x), f 0 (x), . . . , f (N ) (x)), G(x) = (g(x), g 0 (x), . . . , g (N ) (x)). Then,
(α)
for every α ∈ R, the monic generalized Laguerre polynomials {Ln }n are
(N,α+N )
orthogonal with respect to (., .)S
with N = max{0, [−α]}, ( [α] denotes
the greatest integer less than or equal to α).
(N,α+N )
In the case when α is a negative integer, the inner product (., .)S
is the same as the one considered by Kwon and Littlejohn.
The above results justify the interest to consider such a kind of inner
products. In a more general setting, our aim is to study polynomials which
are orthogonal with respect to a symmetric bilinear form such as

(N )



BS (f, g) = f (c), f 0 (c), . . . , f (N −1) (c) A 
g(c)
g 0 (c)
..
.



 + hu, f (N ) g (N ) i,

g (N −1) (c)
(1.1)
where u is a quasi–definite (or regular) linear functional on the linear space
P of real polynomials, c is a real number, N is a positive integer number,
and A is a symmetric N × N real matrix such that each of its principal
submatrices is regular. By analogy with the usual terminology, we call it a
discrete–continuous Sobolev bilinear form. Recently some properties of the
(1)
polynomials orthogonal with respect to BS (., .) had been considered in [4].
We will emphasize some cases in which the functional u satisfies some
extra conditions, namely, u is a semiclassical or a classical linear functional
(see [3], [7] and [9]). A quasi–definite linear functional u is called semiclassical if there exist polynomials φ and ψ with degφ ≥ 0 and degψ ≥ 1 such that
u satisfies the distributional differential equation D(φu) = ψu. Whenever
degφ ≤ 2 and degψ = 1, the functional u is called classical. It is well known
that the only classical functionals correspond to the sequences of Hermite,
Laguerre, Jacobi and Bessel polynomials.
In Section 2, we give a description of the monic polynomials {Qn }n which
(N )
are orthogonal with respect to BS (., .) in terms of the monic polynomials
3
{Pn }n orthogonal with respect to the functional u. In particular, for n ≥ N ,
(N )
(k)
n!
we have that Qn (x) = (n−N
)! Pn−N (x) and Qn (c) = 0 for k = 0, 1, ..., N −
−1
1, while {Qn }N
n=0 are orthogonal with respect to the discrete part of the
symmetric bilinear form (1.1) and they are determined by the matrix A.
By using these results, in Section 3, we give some examples of polynomials orthogonal with respect to (1.1), with an adequate choice of c, namely,
(−N )
(−N,β)
Laguerre polynomials {Ln }n with c = 0, Jacobi polynomials {Pn
}n
(α,−N )
with c = 1, β +N not being a negative integer, and {Pn
}n with c = −1,
α+N not being a negative integer. Note that these sequences of polynomials
are not orthogonal with respect to any quasi–definite linear functional.
In Section 4, we give a new characterization of classical polynomials as
the only orthogonal polynomials such that, for some positive integer number
N , they have a N –th primitive satisfying a three–term recurrence relation.
In particular, this result is applied to discrete–continuous Sobolev polynomials which satisfy a three–term recurrence relation and then it follows that
u is classical with distributional differential equation D(φu) = ψu, and the
point c in (1.1) is such that φ(c) = 0 and ψ(c) = φ0 (c). Hence, the only
monic discrete–continuous Sobolev polynomials which satisfy a three–term
recurrence relation are the ones described in Section 3.
The link between Sobolev orthogonality and polynomials satisfying a
second order differential equation is analyzed in Section 5. It is proved that
if the sequence {Qn }n satisfies the equation
φ(x)Q00n (x) + σ(x)Q0n (x) = ρn Qn (x),
where φ and σ are polynomials with degree less than or equal to 2 and
1, respectively, and ρn are real numbers, then the functional u is classical
with distributional differential equation D(φu) = ψu, σ(x) = ψ(x) − N φ0 (x)
and the point c in (1.1) verifies φ(c) = 0 and ψ(c) = φ0 (c). Hence, the only
monic discrete–continuous Sobolev polynomials which satisfy a second order
differential equation are again the described in Section 3.
As a consequence of the results in Sections 4 and 5, we have that if u is not
a classical linear functional then the sequence {Qn }n does not satisfy neither
a three–term recurrence relation nor a second order differential equation. In
order to avoid this lack in our study, in Section 6, we introduce a linear
differential operator F (N ) on P symmetric with respect to the bilinear form
(1.1). The basic property of this operator is a relationship between the
Sobolev bilinear form and the bilinear form associated with the functional
4
u. Handling with F (N ) we can deduce explicit relations between {Qn }n
and {Pn }n as well as a differential substitute of the algebraic recurrence
relations. This is done in Section 7.
2
The Sobolev discrete-continuous bilinear form
Let P be the linear space of real polynomials, u a quasi–definite linear functional on P (see [2]), N a positive integer number, and A a quasi–definite
and symmetric real matrix of order N , that is, a symmetric and real matrix
such that all the principal minors are different from zero. For a given real
number c, the expression

(N )



BS (f, g) = f (c), f 0 (c), . . . , f (N −1) (c) A 
g(c)
g 0 (c)
..
.



 + hu, f (N ) g (N ) i,

g (N −1) (c)
(2.1)
defines a symmetric bilinear form on P.
Since expression (2.1) involves derivatives, this bilinear form is non–
standard, and by analogy with the usual terminology we will call it a
discrete–continuous Sobolev bilinear form.
In the linear space of real polynomials, we can consider the basis given
by
(x − c)m
.
(2.2)
m!
m≥0
For n ≤ N − 1, the associated Gram matrix Gn is given by the n–th order
principal submatrix of the matrix A. For n ≥ N , the associated Gram
matrix is given by
!
A
0
Gn =
,
0 Bn−N
where Bn−N is the Gram matrix associated with the quasi–definite linear
functional u in the basis (2.2).
In both cases, Gn is quasi–definite (that is, all the principal minors are
different from zero) and therefore, we will say that the discrete–continuous
Sobolev bilinear form (2.1) is quasi–definite. Thus, we can assure the existence of a sequence of monic polynomials, denoted by {Qn }n , which are
orthogonal with respect to (2.1). These polynomials will be called Sobolev
orthogonal polynomials. Our first aim is to relate this sequence with the
5
monic orthogonal polynomial sequence (in short MOPS) {Pn }n associated
with the quasi–definite linear functional u.
Theorem 2.1 Let {Qn }n be the sequence of monic orthogonal polynomials
(N )
with respect to the bilinear form BS .
N −1
i) The polynomials {Qn }n=0
are orthogonal with respect to the discrete bilinear form
g(c)
g 0 (c)
..
.

(N )



BD (f, g) = f (c), f 0 (c), . . . , f (N −1) (c) A 



.

(2.3)
g (N −1) (c)
ii) If n ≥ N , then
Qn(k) (c) = 0,
)
Q(N
n (x) =
k = 0, 1, . . . , N − 1,
n!
Pn−N (x).
(n − N )!
(N )
(2.4)
(2.5)
(N )
Proof. i) If 0 ≤ m, n < N , then Qn (x) = Qm (x) = 0, and the value
of the Sobolev bilinear form on (Qn , Qm ) can be computed by means of the
following expression
(N )
(N )
BS (Qn , Qm ) = BD (Qn , Qm )

=



−1)
Qn (c), Q0n (c), . . . , Q(N
(c) A 
n
Qm (c)
Q0m (c)
..
.
(N −1)
Qm



,

(c)
and therefore they are orthogonal with respect to the discrete bilinear form
(2.3).
ii) Let n ≥ N , then from the orthogonality of the polynomial Qn , we deduce
0
 
 ... 
 
1
(N )
 
−1)
0 = BS (Qn (x), (x − c)k ) = Qn (c), Q0n (c), . . . , Q(N
(c) A  1  ,
n
.
k!
 .. 
0
(2.6)
6
for 0 ≤ k ≤ N − 1. Thus, the vector
−1)
Qn (c), Q0n (c), . . . , Q(N
(c)
n
is the only solution of a homogeneous linear system with N equations and
N unknowns, whose coefficient matrix A is regular. Then, we conclude
(k)
Qn (c) = 0, k = 0, 1, . . . , N − 1, that is, Qn contains the factor (x − c)N .
(N )
In this way, if n, m ≥ N , the discrete part of the bilinear form BS (Qn , Qm )
vanishes and we get
(N )
) (N )
BS (Qn , Qm ) = hu, Q(N
n Qm i = k̃n δn,m ,
k̃n 6= 0.
(N )
That is, the polynomials {Qn }n≥N are orthogonal with respect to the
linear functional u, and equality (2.5) follows from a simple inspection of
the leading coefficients.
Reciprocally, we are going to show that a system of monic polynomials
{Qn }n satisfying equations (2.4) and (2.5) is orthogonal with respect to some
discrete–continuous Sobolev bilinear form. This result could be considered
a Favard–type theorem.
Theorem 2.2 Let {Pn }n be the MOPS associated with a quasi–definite linear functional u, and N ≥ 1 a given integer number. Let {Qn }n be a sequence of monic polynomials satisfying
i) deg Qn = n, n = 0, 1, 2, . . . ,
(k)
ii) Qn (c) = 0, 0 ≤ k ≤ N − 1, n ≥ N,
n!
(N )
iii) Qn (x) =
Pn−N (x), n ≥ N.
(n − N )!
Then, there exists a quasi–definite and symmetric real matrix A, of order
N , such that {Qn }n is the monic orthogonal polynomial sequence associated
with the Sobolev bilinear form defined by (2.1).
Proof. By using the same reasoning as above it is obvious that every
polynomial Qn , with n ≥ N , is orthogonal to every polynomial with degree
less than or equal to n − 1 with respect to a Sobolev bilinear form like (2.1)
containing an arbitrary matrix A in the discrete part and the functional u
in the second part.
Next, we show that we can recover the matrix A from the N first polynomials Qk , k = 0, 1, . . . , N − 1.
7
Let us denote



Q=


Q00 (c)
Q01 (c)
..
.
Q0 (c)
Q1 (c)
..
.
...
...
QN −1 (c) Q0N −1 (c) . . .
(N −1)

Q0
(c)

(N −1)
Q1
(c) 
,

..

.
(N −1)
QN −1 (c)
then Q is a lower triangular and invertible matrix. Let D be a diagonal
matrix with non zero elements in its diagonal.
Define
A = Q−1 D(Q−1 )T .
Obviously A is symmetric and quasi–definite and since
QAQT = D,
the polynomials Q0 , . . . , QN −1 are orthogonal with respect to the bilinear
form (2.1), with the matrix A in the discrete part. Besides, the elements in
(N )
the diagonal of D are the values BS (Qk , Qk ) for k = 0, . . . , N − 1.
Remark. Observe that the matrix A is not unique, because its construction
depends on the arbitrary regular diagonal matrix D.
3
Classical examples
3.1
The Laguerre case
Let α ∈ R, the n–th monic generalized Laguerre polynomial is defined in
[12], p. 102, by means of its explicit representation
n
L(α)
n (x) = (−1) n!
n
X
(−1)j n + α
j=0
j!
n−j
xj ,
n ≥ 0,
(3.1)
where
a
k
denotes the generalized binomial coefficient
a
k
=
(a − k + 1)k
,
k!
(3.2)
and (a − k + 1)k stands for the so-called Pochhammer’s symbol defined by
(b)0 = 1,
(b)n = b(b + 1) . . . (b + n − 1),
8
for b ∈ R,
n ≥ 0.
(3.3)
In this way, we have
(k)
(L(α)
n ) (0)
n+k
= (−1)
n!
n+α
n−k
,
n ≥ k.
If α is a negative integer number, say α = −N , for n ≥ N , we have
) (k)
(L(−N
) (0) = 0,
n
k = 0, 1, . . . , N − 1,
and, for n < N , we get
) (k)
(L(−N
) (0)
n
= n!
N −k−1
n−k
,
k = 0, 1, . . . , n.
On the other hand, since the derivatives of Laguerre polynomials are again
Laguerre polynomials, we have
) (N )
(L(−N
) (x) =
n
n!
(0)
L
(x),
(n − N )! n−N
for n ≥ N.
Therefore, from the previous Section, we conclude that Laguerre polynomials
(−N )
Ln
are orthogonal with respect to the Sobolev bilinear form
(N )
BS (f, g)
Z
T
= F (0)AG(0) +
+∞
f (N ) (x)g (N ) (x)e−x dx,
0
with F (0) = (f (0), f 0 (0), . . . , f (N −1) (0)), G(0) = (g(0), g 0 (0), . . . , g (N −1) (0)),
the matrix A is given by
A = Q−1 D(Q−1 )T ,
(−N )
Q is the matrix of the derivatives of Laguerre polynomials Ln
evaluated
at zero


N −1
0!
0
...
0


0 

N −1
N −2




1!
1!
.
.
.
0


1
0
Q=



..
..
..


.
.
.

N −1
N −2
0 
(N − 1)!
(N − 1)!
. . . (N − 1)!
N −1
N −2
0
and D is an arbitrary regular diagonal matrix. Similar results have been
obtained with different techniques in [5] and [10]. We recover the results
in [5] by using a diagonal matrix D whose elements are (0!)2 , (1!)2 , . . . ,
((N − 1)!)2 .
9
3.2
The Jacobi case
For α and β arbitrary real numbers, the generalized Jacobi polynomials can
be defined (see [12], p. 62) by means of their explicit representation
Pn(α,β) (x) =
n X
x − 1 n−m x + 1 m
n+α
n+β
m=0
n−m
m
2
2
,
n ≥ 0.
When α, β and α + β + 1 are not a negative integer, Jacobi polynomials are
orthogonal with respect to the quasi–definite linear functional u(α,β) . This
linear functional is positive definite for α > −1 and β > −1.
For α = −N , with N a positive integer, and β being not a negative
integer, the n–th monic generalized Jacobi polynomial is given by
Pn(−N,β) (x)
=
2n − N + β
n
−1 X
n n−N
n+β
n−m
m
m=0
(x − 1)n−m (x + 1)m . (3.4)
In this case, for n ≥ N , x = 1 will be a zero of multiplicity N ([12], p. 65).
On the other hand, since the derivatives of Jacobi polynomials are again
Jacobi polynomials, we have
(Pn(−N,β) )(N ) (x) =
n!
(0,β+N )
P
(x),
(n − N )! n−N
for n ≥ N.
Therefore, from the previous Section, we conclude that Jacobi polynomials
(−N,β)
Pn
, when β + N is not a negative integer, are orthogonal with respect
to the Sobolev bilinear form
(N )
BS (f, g) = F (1)AG(1)T + hu(0,β+N ) , f (N ) g (N ) i,
where the matrix A is given by
A = Q−1 D(Q−1 )T ,
Q is the matrix of the derivatives of Jacobi polynomials
Q = (Pn(−N,β) )(k) (1)
n,k=0,...,N −1
which are given by
(Pn(−N,β) )(k) (1) = 2n−k
(−N + k + 1)n−k
n!
,
(n − k)! (n − N + β + k + 1)n−k
and D is an arbitrary regular diagonal matrix.
Of course, a similar result can be stated in the case when α + N is not
a negative integer, β = −N , and c = −1.
10
4
Sobolev orthogonal polynomials and three-term
recurrence relations
Laguerre and Jacobi polynomials satisfy a three–term recurrence relation
even for negative integer values of their respective parameters (see [12]). In
the previous Section, we have seen that Laguerre polynomials with α a negative integer and Jacobi polynomials with either α or β a negative integer
are Sobolev orthogonal polynomials. In this way a natural question arises:
do the Sobolev orthogonal polynomials satisfy a three–term recurrence relation? As we are going to show, the answer is very restrictive, the existence
of a three–term recurrence relation for the Sobolev orthogonal polynomials
implies the classical character of the linear functional u associated with the
bilinear form (2.1).
Definition 4.1 We will say that a family of polynomials {Qn }n≥0 is a
monic polynomial system (MPS) if
i) deg(Qn ) = n,
ii) Q0 (x) = 1,
n ≥ 0,
Qn (x) = xn + lower degree terms,
n ≥ 1.
Obviously, every MPS is a basis of the linear space P and every MOPS
is a MPS.
Definition 4.2 A monic polynomial system {Qn }n≥0 satisfies a three–term
recurrence relation if there exist two sequences of real numbers {bn }∞
n=0 and
{gn }∞
,
such
that
n=1
xQn (x) = Qn+1 (x) + bn Qn (x) + gn Qn−1 (x),
Q−1 (x) = 0,
n ≥ 0,
Q0 (x) = 1.
From Favard’s theorem (see [2], p. 21) we can deduce the existence of
monic polynomial systems satisfying a three–term recurrence relation which
are not orthogonal with respect to any linear functional. This case appears
when some of the coefficients gn are zero. For instance, Laguerre polynomials
with parameter α a negative integer and Jacobi polynomials with parameters
either α or β or α + β + 1 a negative integer.
11
Proposition 4.3 Let {Qn }n≥0 be a monic polynomial system satisfying a
three–term recurrence relation and let N be a positive integer number such
that the system of monic N –th order derivatives
Pn (x) :=
n!
(N )
Q
(x),
(n + N )! n+N
n ≥ 0,
constitutes a monic orthogonal polynomial sequence. Then, the polynomials
{Pn }n are classical.
Proof. Since {Pn }n≥0 is a MOPS, it satisfies a three–term recurrence relation
xPn (x) = Pn+1 (x) + βn Pn (x) + γn Pn−1 (x), n ≥ 0,
P−1 (x) = 0,
P0 (x) = 1,
with γn 6= 0, n ≥ 0.
In this way,
n+1
n + N (N )
(N )
(N )
Qn+N +1 (x) + βn Qn+N (x) + γn
Qn+N −1 (x).
n+N +1
n
(4.1)
On the other hand, the monic polynomial sequence {Qn }n≥0 satisfies a
three–term recurrence relation
(N )
xQn+N (x) =
xQn+N (x) = Qn+N +1 (x) + bn+N Qn+N (x) + gn+N Qn+N −1 (x).
Taking N –th order derivatives in this relation, we get
(N −1)
(N )
(N )
(N )
(N )
xQn+N (x) + N Qn+N (x) = Qn+N +1 (x) + bn+N Qn+N (x) + gn+N Qn+N −1 (x).
(4.2)
(N )
By eliminating the term xQn+N (x) between (4.1) and (4.2), we obtain
(N −1)
N Qn+N (x) =
N
(N )
Q
+1 (x)+
n + N + 1 n+N
(N )
+ (bn+N − βn ) Qn+N (x) + gn+N −
n+N
(N )
γn Qn+N −1 (x). (4.3)
n
Taking again derivatives in this relation we deduce that each polynomial
Pn can be expressed as a linear combination of the derivatives of three
consecutive polynomials in the sequence {Pn }n and, therefore, we conclude
that they are classical by using the characterization of classical orthogonal
polynomials obtained by Marcellán et al. in [7].
12
Remark. This result characterizes the classical orthogonal polynomials as
the only system of orthogonal polynomials having a N –th order primitive
(N ≥ 1) which satisfies a three–term recurrence relation.
Theorem 4.4 Let {Qn }n be the monic orthogonal polynomial sequence associated with the Sobolev bilinear form (2.1). If the polynomials {Qn }n satisfy a three-term recurrence relation, then the linear functional u is classical
and the point c in (2.1) satisfies
φ(c) = 0,
(4.4)
ψ(c) − φ0 (c) = 0,
(4.5)
where φ and ψ are the polynomials in the distributional differential equation
D(φu) = ψu satisfied by u.
Proof. Let {Pn }n be the monic orthogonal polynomial sequence associated
with the linear functional u. From Theorem 2.1, we have
Q(k)
n (c) = 0,
)
Q(N
n (x) =
k = 0, 1, . . . , N − 1,
n!
Pn−N (x),
(n − N )!
(4.6)
(4.7)
for all n ≥ N . Therefore, using Proposition 4.3, we deduce the classical
character of the polynomials {Pn }n and then the classical linear functional
u satisfies a distributional differential equation
D(φu) = ψu,
where φ and ψ are polynomials with deg φ ≤ 2 and deg ψ = 1. From
Bochner’s characterization of the classical orthogonal polynomials, (see [2]),
we deduce that the polynomials {Pn }n satisfy the second order differential
equation
φ(x)Pn00 (x) + ψ(x)Pn0 (x) = λn Pn (x),
for all n ≥ 0.
Thus the polynomials {Qn }n satisfy the differential equation
(N +2)
(N +1)
(N )
φ(x)Qn+N (x) + ψ(x)Qn+N (x) = λn Qn+N (x),
13
for n ≥ 0. This differential equation can be written in a more convenient
way
(N +1)
0
φ(x)Qn+N (x)
0
(N )
+ (ψ(x) − φ0 (x))Qn+N (x)
(N )
= κn Qn+N (x),
(4.8)
where κn = λn + ψ 0 (x) − φ00 (x). Integrating (4.8) we get
(N +1)
(N −1)
(N )
φ(x)Qn+N (x) + (ψ(x) − φ0 (x))Qn+N (x) = κn Qn+N (x) + µn ,
where µn is a constant.
For n ≥ 2, let p be a polynomial with deg p ≤ n − 2, then
h
(N +1)
(N )
i
(N +1)
(N )
hu, p φQn+N + (ψ − φ0 )Qn+N i = hu, pφQn+N i − hu, (φpQn+N )0 i
(N )
= −hu, (pφ)0 Qn+N i
(n + N )!
= −
hu, (pφ)0 Pn i = 0.
n!
(N +1)
(N )
(N −1)
Thus, the polynomial φQn+N + (ψ − φ0 )Qn+N = κn Qn+N + µn is orthogonal, with respect to u, to every polynomial of degree less than or equal to
n−2, and then it can be written as a linear combination of three consecutive
polynomials Pn
(N −1)
κn Qn+N + µn = κn Pn+1 + sn Pn + tn Pn−1 .
(N −1)
¿From (4.3) we have that the polynomial Qn+N is a linear combination of
the three polynomials Pn+1 , Pn and Pn−1 , and, since the sequence {Pn }n
constitutes a basis of the linear space of the polynomials, we conclude that
µn = 0, for n ≥ 2.
In this way, the polynomials {Qn+N }n satisfy the differential equation
(N +1)
(N )
(N −1)
φ(x)Qn+N (x) + (ψ(x) − φ0 (x))Qn+N (x) = κn Qn+N (x),
(4.9)
for n ≥ 2.
Replacing x = c in (4.9), from (4.6) we conclude
φ(c)Pn0 (c) + (ψ(c) − φ0 (c))Pn (c) = 0,
for n ≥ 2.
¿From recurrence relation
xPn (x) = Pn+1 (x) + βn Pn (x) + γn Pn−1 (x),
14
(4.10)
satisfied by {Pn }n and (4.10) written for n + 1, n and n − 1, we obtain
cφ(c)Pn0 (c) + φ(c) + c(ψ(c) − φ0 (c)) Pn (c) = 0,
n ≥ 3,
(4.11)
and subtracting (4.10) from (4.11), we get
n ≥ 3.
φ(c)Pn (c) = 0,
Therefore, we conclude φ(c) = 0 and using again (4.10), ψ(c) − φ0 (c) = 0.
Corollary 4.5 The only sequences of monic polynomials which are orthogonal with respect to a Sobolev bilinear form (2.1) and satisfy a three–term
recurrence relation are
(−N )
a) The generalized Laguerre polynomials Ln
,
(−N,β)
, with β + N not a negative
(α,−N )
, with α+N not a negative
b) The generalized Jacobi polynomials Pn
integer,
c) The generalized Jacobi polynomials Pn
integer.
Proof. Suppose that the monic polynomials {Qn }n orthogonal with respect
to (2.1) satisfy a three–term recurrence relation. Theorem 4.4 assures that u
is a classical linear functional. If D(φu) = ψu is its distributional differential
equation, the polynomials φ and ψ are given by the following table
Name
Hermite
Laguerre
Jacobi
φ
1
x
1 − x2
ψ
−2x
(α + 1) − x
(β − α) − (α + β + 2)x
Bessel
x2
(α + 2)x + 2
Restrictions
α 6= −n, n ≥ 1
α 6= −n, β 6= −n,
α + β + 1 6= −n, n ≥ 1
α 6= −n, n ≥ 2
Then, conditions (4.4) and (4.5) exclude Hermite and Bessel cases. Moreover, in Laguerre and Jacobi cases the only possibilities are the following:
- Laguerre case with α = 0 and c = 0.
- Jacobi case with α = 0, β 6= −m, m ≥ 1 and c = 1.
- Jacobi case with β = 0, α 6= −m, m ≥ 1 and c = −1.
Taking into account the results of Section 3, we conclude.
15
(α,β)
Note that a reduction of the degree of Pn
could occur when either
both α and β + n or α + β + 1 are negative integers (see [12], p.64). An
interesting open problem is to give some kind of orthogonality relations valid
for these polynomials. For the particular case α = β negative integers, this
problem has been solved in [1]: the corresponding Gegenbauer polynomials
are orthogonal with respect to a discrete-continuous Sobolev bilinear form,
where the discrete part is concentrated in two points, namely, 1 and −1.
Corollary 4.6 The monic orthogonal polynomials associated to the Sobolev
bilinear form (2.1) satisfy a three term recurrence relation if and only if the
linear functional u is classical and the point c in (2.1) satisfies φ(c) = 0 and
ψ(c) = φ0 (c).
Proof. It follows from Theorem 4.4 and Corollary 4.5 taking in mind that
Laguerre and Jacobi polynomials satisfy a three–term recurrence relation
for all values of their parameters.
5
Sobolev orthogonal polynomials and second order differential equations
As it is well known (see [12]) Laguerre and Jacobi polynomials satisfy a
second order differential equation for every value of their respective parameters.
In this Section, our aim is to characterize the sequences of monic Sobolev
orthogonal polynomials satisfying a second order differential equation. We
can observe that if, for every n, the polynomials {Qn }n satisfy a second
order differential equation
φ(x)Q00n (x) + σ(x)Q0n (x) = ρn Qn (x),
where φ and σ are fixed polynomials and ρn ∈ R, then
deg φ ≤ 2,
deg σ ≤ 1.
Moreover, if ρ1 6= 0 then deg σ = 1.
16
Theorem 5.1 Let {Qn }n be the monic orthogonal polynomial sequence associated with the Sobolev bilinear form (2.1). If, for n ≥ N , every polynomial
Qn satisfies a second order differential equation
φ(x)Q00n (x) + σ(x)Q0n (x) = ρn Qn (x),
(5.1)
where φ and σ are fixed polynomials with degree less than or equal to 2 and
1 respectively, and ρn ∈ R, then the linear functional u is classical with
distributional differential equation D(φu) = ψu, σ(x) = ψ(x) − N φ0 (x) and
the point c in (2.1) satisfies
φ(c) = 0,
(5.2)
(N − 1)φ0 (c) + σ(c) = 0.
(5.3)
Proof. Taking k–th order derivatives in (5.1), from Leibniz rule, we get
φ(x)Q(k+2)
(x) + kφ0 (x) + σ(x) Q(k+1)
(x)
n
n
k(k − 1) 00
=
ρn −
φ (x) − kσ 0 (x) Q(k)
n (x).
2
(5.4)
Let k = N in (5.4), then we deduce that the polynomials Pn orthogonal
with respect to the linear functional u satisfy the second order differential
equation
φ(x)Pn00 (x) + ψ(x)Pn0 (x) = λn Pn (x), n ≥ 0,
where
ψ(x) = N φ0 (x) + σ(x),
N (N − 1) 00
φ (x) − N σ 0 (x).
λn = ρn+N −
2
Therefore u satisfies D(φu) = ψu and deg ψ ≥ 1. Now, since deg φ ≤ 2 and
deg σ ≤ 1, from Bochner’s characterization of the classical orthogonal polynomials, (see [2]), we deduce the classical character of the linear functional
u.
Writing equation (5.4) for n = N and k = N − 1 we get
(N )
(N − 1)φ0 (x) + σ(x) QN (x)
(N − 1)(N − 2) 00
(N −1)
0
=
ρN −
φ (x) − (N − 1)σ (x) QN
(x),
2
17
(N −1)
and by substitution in c, since QN
(c) = 0, we deduce
(N − 1)φ0 (c) + σ(c) = 0.
Finally, if N = 1 writing (5.1) for n = 2, we get φ(c) = 0 and when N ≥ 2,
writing equation (5.4) for n = N , k = N − 2, we get
(N −1)
(N )
φ(x)QN (x) + ((N − 2)φ0 (x) + σ(x))QN
(x)
(N − 2)(N − 3) 00
(N −2)
ρN −
=
φ (x) − (N − 2)σ 0 (x) QN
(x)
2
and by substitution in c we deduce φ(c) = 0.
Using the same reasoning as in Corollary (4.5), we obtain
Corollary 5.2 The only sequences of monic polynomials which are orthogonal with respect to a Sobolev bilinear form (2.1) and satisfy a second order
differential equation (5.1) are
(−N )
a) The generalized Laguerre polynomials Ln
,
(−N,β)
, with β + N not a negative
(α,−N )
, with α+N not a negative
b) The generalized Jacobi polynomials Pn
integer,
c) The generalized Jacobi polynomials Pn
integer.
Corollary 5.3 The monic orthogonal polynomials associated to the Sobolev
bilinear form (2.1) satisfy a second order differential equation like (5.1) if
and only if the linear functional u is classical and the point c in (2.1) satisfies
φ(c) = 0 and ψ(c) = φ0 (c).
Proof. It follows from Theorem 5.1 and Corollary 5.2 taking in mind that
Laguerre and Jacobi polynomials satisfy a second order differential equation
for all values of their parameters.
18
6
A symmetric differential operator: Properties
In order to obtain explicit relations between the sequences {Qn }n and {Pn }n
(N )
associated with BS and u, respectively, we introduce a linear differential
operator F (N ) closely related to u. To do this, u must satisfy an extra
condition. This is why, from now on, the functional u in (2.1) will be a
semiclassical one.
Definition 6.1 ([3], [9]) A linear functional u on P is called semiclassical,
if there exist two polynomials φ and ψ, with deg φ = p ≥ 0 and deg ψ = q ≥
1, such that u satisfies the following distributional differential equation
D(φu) = ψu,
(6.1)
φDu = (ψ − φ0 )u.
(6.2)
or equivalently
Equation (6.2) can be generalized in the following way:
Lemma 6.2 ([8]) Let u be a semiclassical linear functional, then for every
n we have
φn (x)Dn u = ψ(x; n)u,
(6.3)
where the polynomials ψ(x; n) are recursively defined by
ψ(x; 0) = 1,
ψ(x; n) = φ(x)ψ 0 (x; n − 1) + ψ(x; n − 1)[ψ(x) − nφ0 (x)],
n ≥ 1. (6.4)
Observe that, now, (6.2) adopts the form φ(x)Du = ψ(x; 1)u. Taking
derivatives (n − 1)–times in this formula, we get
φ(x)Dn u =
n−1
X i=0
n−1
i
Dn−1−i ψ(x; 1) −
n−1
i−1
Dn−i φ(x) Di u, (6.5)
n
= 0 whenever m < 0. Multiplying by φn−1 and
m
using (6.3), we obtain another recursive expression for ψ(x; n):
for n ≥ 1, where
ψ(x; n)
=
n−1
X i=0
n−1
i
D
n−1−i
ψ(x; 1) −
n−1
i−1
D
n−i
φ(x) φn−1−i (x)ψ(x; i)
(6.6)
19
valid for n ≥ 1.
Lemma 6.3 In the above conditions, we have
φn−j (x)ψ(x; j)Dn u = ψ(x; n)Dj u,
n ≥ j,
(6.7)
and consequently
φi (x)ψ(x; N − i)DN −j u = φj (x)ψ(x; N − j)DN −i u,
0 ≤ i, j ≤ N. (6.8)
Proof. We will show the result by induction on n. The case n = 1, j = 0
comes from (6.3), while the case n = j = 1 is trivial. Suppose that
φn−1−j (x)ψ(x; j)Dn−1 u = ψ(x; n − 1)Dj u,
holds for all j, 0 ≤ j ≤ n − 1. Then,
i) For 0 ≤ j < n − 1, taking derivatives, multiplying by φ, using (6.4), and
the induction hypothesis, we have
φn−j (x)ψ(x; j)Dn u = φ(x)ψ 0 (x, n − 1)Dj u + φ(x)ψ(x; n − 1)Dj+1 u−
−{(n − 1 − j)φ0 (x)ψ(x; j) + ψ(x; j + 1) −
−ψ(x; j)[ψ(x) − (j + 1)φ0 (x)]}φn−1−j (x)Dn−1 u
= φ(x)ψ 0 (x, n − 1)Dj u + φ(x)ψ(x; n − 1)Dj+1 u −
−{ψ(x; j + 1) − ψ(x; j)[ψ(x) − nφ0 (x)]}φn−1−j (x)Dn−1 u
= {φ(x)ψ 0 (x, n − 1) + ψ(x; n − 1)[ψ(x) − nφ0 (x)]}Dj u +
+φ(x)ψ(x; n − 1)Dj+1 u − ψ(x; j + 1)φn−1−j (x)Dn−1 u
= ψ(x; n)Dj u + φ(x)ψ(x; n − 1)Dj+1 u − φ(x)ψ(x; n − 1)Dj+1 u
= ψ(x; n)Dj u.
ii) If j = n − 1, multiplying (6.5) by ψ(x; n − 1), and using the induction
hypothesis, we obtain the result taking into account (6.6).
Now, from i) and ii), we conclude, since the case j = n is trivial.
We define a linear differential operator F (N ) on the linear space of real
polynomials P in the following way
F
(N )
N
N
= (−1) (x − c)
N X
N
i
i=0
20
φi (x)ψ(x; N − i)DN +i ,
(6.9)
where D denotes the derivative operator and the polynomials ψ(x; n) are
defined as in Lemma 6.2.
Remark. In the particular case of a semiclassical linear functional u defined
from a weight function, expression (6.9) can be written in a very compact
form
φN (x) N ρ(x)DN
F (N ) = (−1)N (x − c)N
ρ(x)
where ρ denotes the weight function associated with the semiclassical linear
functional u.
In the next Lemma, we recall a very useful formula involving derivatives.
Lemma 6.4 ([8]) Let f and g be n–times and 2n–times differentiable functions, respectively. Then,
f
(n) (n)
g
=
n
X
i
(−1)
i=0
n
i
f g (n+i)
(n−i)
.
In the following Proposition, we show how the linear operator F (N ) allows us to obtain a representation for the Sobolev bilinear form (2.1), in
terms of the consecutive derivatives of the semiclassical linear functional u.
(N )
Proposition 6.5 Let BS be a Sobolev bilinear form with u semiclassical
and f , g arbitrary polynomials. Then, for 0 ≤ i ≤ N , we have
(N )
BS
(x − c)N φi (x)ψ(x; N − i)f, g = hDN −i u, f F (N ) gi.
Proof. From Lemmas 6.2, 6.3 and 6.4, we get
(N )
BS
(x − c)N φi (x)ψ(x; N − i)f, g
= hu, (x − c)N φi (x)ψ(x; N − i)f
=
N
X
j
(−1)
j=0
=
N
X
j=0
(−1)N
N
j
N
j
(N )
g (N ) i
hu, (x − c)N φi (x)ψ(x; N − i)f g (N +j)
(N −j)
hDN −j u, (x − c)N φi (x)ψ(x; N − i)f g (N +j) i
21
i
=
N
X
(−1)N
N
j
hφi (x)ψ(x; N − i)DN −j u, (x − c)N f g (N +j) i
N
j
hφj (x)ψ(x; N − j)DN −i u, (x − c)N f g (N +j) i
j=0
=
N
X
N
(−1)
j=0
= hDN −i u, f [(−1)N (x − c)N
N X
N
j=0
= hD
N −i
u, f F
(N )
j
φj (x)ψ(x; N − j)g (N +j) ]i
gi.
Theorem 6.6 The linear operator F (N ) is symmetric with respect to the
Sobolev bilinear form (2.1), that is
(N )
(N )
BS (F (N ) f, g) = BS (f, F (N ) g).
Proof. From Proposition 6.5 and Lemma 6.4, we can deduce
(N )
BS (F (N ) f, g) =
N
X
(−1)N
i=0
=
=
N
X
i=0
N
X
(−1)N
(−1)i
i=0
N
i
N
i
N
i
(N )
BS
(x − c)N φi (x)ψ(x; N − i)f (N +i) , g
hDN −i u, f (N +i) F (N ) gi
hu, f (N +i) F (N ) g
= hu, f (N ) F (N ) g
(N )
(N −i)
i
(N )
i = BS (f, F (N ) g).
Now, we study the degree of the polynomial F (N ) xn for every n. Observe
that F (N ) vanishes on every polynomial with degree less than N .
Proposition 6.7 For every n ≥ 0, we have
deg F (N ) xn ≤ n + N max{p − 1, q},
where p = deg φ and q = deg ψ.
22
Proof. By using the induction method it is very easy to see that deg ψ(x; n)
≤ n + max{p − 1, q} for all n ≥ 0. Taking into account the definition of the
linear operator F (N ) , the conclusion follows.
On the other hand, the linear operator F (N ) never reduces the degree
for all polynomials.
Proposition 6.8 There exists n0 ≥ N such that
deg F (N ) xn0 ≥ n0 .
Proof. Suppose that deg F (N ) xn < n, for all n ≥ N . Then, we can expand
F (N ) Qn =
n−1
X
an,i Qi .
i=0
Thus, since F (N ) vanishes on every polynomial of degree less than N , using
its symmetry property, we have
(N )
an,i =
BS
F (N ) Qn , Qi
(N )
=
(N )
BS
Qn , F (N ) Qi
(N )
BS (Qi , Qi )
= 0,
i = 0, 1, . . . , n − 1,
BS (Qi , Qi )
and the result follows.
To study the degree of F (N ) xn , we need to know the degree of the polynomials ψ(x; N − i), i = 0, . . . , N in formula (6.9). The following Lemma
provides us some combinatorial identities, that will be useful for our purpose.
Lemma 6.9 Let a and b arbitrary real numbers. Then for every non negative integer n, we have
i)
ii)
(a)n = (−1)n (−a − n + 1)n ,
n X
a
i
i=0
iii)
n
X
i=0
where
(−1)i
a
k
b
n−i
a
i
=
a+b
,
n
b−i
n−i
=
b−a
,
n
and (a)k are given by (3.2) and (3.3).
23
Proof. i) It is a direct consequence of the definition of the Pochhammer’s
symbol.
ii) It can be derived from the power series expansion of the identity
(1 + x)a (1 + x)b = (1 + x)a+b ,
comparing the coefficients.
iii) This formula can be deduced from i) and ii).
Remark. For a and b positive integer numbers, formulas ii) and iii) can be
found on page 8 in [11].
Let assume that the explicit representation for the polynomials φ and ψ
is given by
φ(x) =
p
X
ai xi ,
ap 6= 0,
p ≥ 0,
ψ(x) =
i=0
q
X
bi xi ,
bq 6= 0,
q ≥ 1,
i=0
and without loss of generality, we can suppose that ap = 1.
Next Lemma gives us the degree of the polynomial ψ(x; n) and its leading
coefficient in terms of p, q and bq .
Lemma 6.10 The following assertions are true:
i) If p − 1 < q, then
ψ(x; n) = bnq xnq + lower degree terms,
and deg ψ(x; n) = nq,
ii) If p − 1 > q, then
n ≥ 0.
ψ(x; n) = (−1)n (p)n xn(p−1) + lower degree terms,
and deg ψ(x; n) = n(p − 1),
iii) If p − 1 = q, then
n ≥ 0.
ψ(x; n) = (−1)n (p − bq )n xn(p−1) + lower degree terms.
Therefore,
iii.1) If p − bq 6= 0, −1, . . . , −(n − 1) then deg ψ(x; n) = n(p − 1), n ≥ 0,
iii.2) If p − bq = −k, k ≥ 0, then deg ψ(x; n) = n(p − 1), 0 ≤ n ≤ k, and
deg ψ(x; n) < n(p − 1), n > k.
24
Proof. These results can be obtained by induction on n.
As we are going to see, the equality in Proposition 6.7 is true for almost
all n, that is, the action of F (N ) on a polynomial of degree bigger than
or equal to N increases its degree exactly in N t being t = max{p − 1, q}.
Therefore, we can write
F (N ) xn = F (n; N, t)xn+N t + . . . ,
where F (n; N, t) denotes the leading coefficient of the polynomial F (N ) xn .
We want to notice that this coefficient can be zero for some specific values
of n and in particular for every n < N .
To prove this, we decompose the operator F (N ) in N + 1 differential
operators defined by
(N )
Fi
N
N
= (−1) (x − c)
N
i
φi (x)ψ(x; N − i)DN +i ,
i = 0, 1, . . . , N.
(6.10)
Thus,
F (N ) =
N
X
min{N,n−N }
(N )
Fi
,
and F (N ) xn =
i=0
X
(N ) n
Fi
x , n ≥ N,
i=0
where, for i = 0, . . . , min{N, n − N },
(N ) n
Fi
x = (−1)N (x − c)N
N
i
φi (x)ψ(x; N − i)
n!
xn−N −i .
(n − N − i)!
(N ) n
x
Let us denote by Fi (n) the leading coefficient of the polynomial Fi
and, for the sake of simplicity, we will put F (n; N, t) = F (n).
Theorem 6.11 Let t = max{p − 1, q}. Except for finitely many values of
n ≥ N , we have
deg F (N ) xn = n + N t,
that is,
F (N ) xn = F (n)xn+N t + lower terms degree,
with F (n) 6= 0. More precisely
i) If p − 1 < q, then
F (n) = (−bq )N
25
n!
,
(n − N )!
and deg F (N ) xn = n + N t,
n ≥ N.
ii) If p − 1 > q, then
F (n) = (p − n + N )N
n!
,
(n − N )!
and
deg F (N ) xn < n + N t,
deg F
(N ) n
x
= n + N t,
N + p ≤ n ≤ 2N − 1 + p,
N ≤ n < N + p,
n ≥ 2N + p.
iii) If p − 1 = q, then
F (n) = (p − bq − n + N )N
n!
,
(n − N )!
and
iii.1) if p − bq = −k,
k = 0, 1, . . . , N − 1, then
deg F (N ) xn < n + N t,
deg F
(N ) n
x
= n + N t,
N ≤ n ≤ 2N − 1 − k,
n ≥ 2N − k,
iii.2) if p − bq is a positive integer, then
deg F (N ) xn < n + N t,
deg F
(N ) n
x
N + p − bq ≤ n ≤ 2N − 1 + p − bq ,
N ≤ n < N + p − bq ,
= n + N t,
n ≥ 2N + p − bq .
iii.3) in another case,
deg F (N ) xn = n + N t,
n ≥ N.
Proof. To prove the theorem a basic tool will be Lemma 6.10. For this
reason, we distinguish three different cases.
i) Case p − 1 < q. In this situation, we have
(N ) n
deg Fi
and then,
x = n + N q − i(q − (p − 1)),
(N ) n
x
deg F (N ) xn = deg F0
26
i = 0, 1, . . . , min{N, n − N },
= n + N q,
for all n ≥ N.
(N ) n
x
The explicit expression for F0
(N ) n
F0
is
= (−1)N (x − c)N ψ(x; N )DN xn
n!
= (−bq )N
xn+N q + lower terms degree,
(n − N )!
x
and the leading coefficient for F (N ) xn is
F (n) = F0 (n) = (−bq )N
ii)
n!
,
(n − N )!
n ≥ N.
Case p − 1 > q. In this case, as p > 2, we get
(N ) n
deg Fi
x = n + N (p − 1),
i = 0, . . . , min{N, n − N },
and then deg F (N ) xn ≤ n + N (p − 1),
(N )
coefficient of Fi xn is
i
Fi (n) = (−1)
N
i
(p)N −i
for all n ≥ N. The leading
n!
,
(n − N − i)!
i = 0, 1, . . . , min{N, n − N }.
Taking into account that
min{N,n−N }
X
F (n) =
Fi (n),
(6.11)
i=0
and using Lemma 6.9 iii), we can show that:
If N ≤ n < 2N , since min{N, n − N } = n − N , we obtain
F (n) =
n−N
X
Fi (n) = n! (p)2N −n
i=0
n−N
X
(−1)i
i=0
= n! (p)2N −n
p−1
n−N
N
i
= (p − n + N )N
p−1+N −i
n−N −i
n!
.
(n − N )!
On the other hand, if n ≥ 2N ,
F (n) =
N
X
Fi (n) =
i=0
=
n! N !
(n − N )!
N
n! N ! X
(−1)i
(n − N )! i=0
p − 1 − n + 2N
N
27
n−N
i
p−1+N −i
N −i
= (p − n + N )N
n!
.
(n − N )!
Observe that, since p is an integer bigger than 2, from the definition of
the Pochhammer’s symbol (p − n + N )N , we have F (n) = 0 if and only if
N + p ≤ n ≤ 2N − 1 + p. Then, there exist exactly N values of n such
that deg F (N ) xn < n + N (p − 1). In another case, we have deg F (N ) xn =
n + N (p − 1).
Case p − 1 = q.
First, we assume that p − bq = −k, k = 0, 1, . . . , N − 1. In this case,
by Lemma 6.10, deg ψ(x; N − i) = (N − i)(p − 1), if 0 ≤ N − i ≤ k and
deg ψ(x; N − i) < (N − i)(p − 1) for k + 1 ≤ N − i ≤ N .
(N )
Therefore, for n ≥ N , we have deg Fi xn < n + N (p − 1) when i =
(N )
0, 1, . . . , N − k − 1, and deg Fi xn = n + N (p − 1) if N − k ≤ i ≤ N . In
(N
)
n
this way, deg F x < n + N (p − 1) when N ≤ n ≤ 2N − k − 1.
iii)
For n ≥ 2N − k, we can observe that
min{N,n−N }
X
F (n) =
Fi (n),
i=N −k
where, by Lemma 6.9 i),
n!
N
(−1)N −i (−k)N −i
Fi (n) = (−1)
i
(n − N − i)!
n!
N
N
= (−1)
(i + 1 − (N − k))N −i
.
i
(n − N − i)!
N
Now, we give an explicit expression of F (n). Suppose 2N − k ≤ n < 2N ,
then min{N, n − N } = n − N , and using Lemma 6.9 ii)
F (n) = (−1)N
n−N
X
i=N −k
N
i
n! N !
= (−1)
(n − N )!
N
(i + 1 − (N − k))N −i
n−(2N −k) X
i=0
k
i
n−N
n − (2N − k) − i
n+k−N
n − (2N − k)
n!
= (−1)N (n + 1 − (2N − k))N
.
(n − N )!
n! N !
= (−1)N
(n − N )!
28
n!
(n − N − i)!
If n ≥ 2N , again by Lemma 6.9 ii), we have
N
F (n) = (−1)
N
X
N
i=N −k
= (−1)N
i
(i + 1 − (N − k))N −i
k
X
n! k!
(n − (2N − k))! i=0
n!
(n − N − i)!
n − (2N − k)
i
N
k−i
n+k−N
k
n!
= (−1)N (n + 1 − (2N − k))N
.
(n − N )!
= (−1)N
n! k!
(n − (2N − k))!
Hence, if n ≥ 2N − k,
F (n) = (−1)N (n + 1 − (2N − k))N
n!
n!
= (−k − n + N )N
6= 0.
(n − N )!
(n − N )!
Now, we assume that p−bq 6= 0, −1, . . . , −(N −1) and thus, from Lemma
6.10, deg ψ(x; N − i) = (N − i)(p − 1), i = 0, 1, . . . , N . As in the case
(N )
p−1 > q, we have deg Fi xn = n+N (p−1), i = 0, 1, . . . , min{N, n−N },
(N
)
n
deg F x ≤ n + N (p − 1), and also (6.11), where
Fi (n) = (−1)i
N
i
(p − bq )N −i
n!
,
(n − N − i)!
i = 0, 1, . . . , min{N, n − N }.
Using the same technique, we obtain
F (n) = (p − bq − n + N )N
n!
,
(n − N )!
n ≥ N.
If p − bq is a positive integer, then F (n) = 0 if and only if N + p − bq ≤
n ≤ 2N − 1 + p − bq , that is, there exist precisely N values of n such that
deg F (N ) xn < n + N (p − 1), and for the other values of n ≥ N , F (n) 6= 0
and hence deg F (N ) xn = n + N (p − 1).
In another case, deg F (N ) xn = n + N (p − 1), for all n ≥ N .
7
Recurrence relations and differential operators
As a direct consequence of Proposition 6.5, which relates the discrete–
(N )
continuous Sobolev bilinear form BS and the one defined from a semiclassical linear functional u, we can establish some relations between the
29
monic Sobolev orthogonal polynomials {Qn }n and the monic orthogonal
polynomials {Pn }n , associated with the semiclassical linear functional u. In
the sequel, for the sake of simplicity, we will denote
(N )
kn = hu, Pn2 i =
6 0,
k̃n = BS (Qn , Qn ) 6= 0,
∀n ≥ 0.
Proposition 7.1 The following formulas hold:
n+N (p+1)
i)
X
(x − c)N φN (x)Pn (x) =
(n)
n ≥ 0,
αi Qi (x),
(7.1)
i=r
(n)
where r = max{N, n−N t},
ii)
αn+N (p+1) = 1
n+N
Xt
F (N ) Qn (x) =
(n)
and
(n)
αr
=
hu, Pn F (N ) Qr i
.
k˜r
n ≥ N (p + 1),
βi Pi (x),
(7.2)
i=n−N (p+1)
(n)
k̃n
.
kn−N (p+1)
(n)
where βn+N t = F (n),
βn−N (p+1) =
Proof.
i) Expanding the polynomial (x − c)N φN Pn in terms of the Sobolev polynomials Qn , we have
n+N (p+1)
(x − c)N φN (x)Pn (x) =
X
(n)
αi Qi (x),
i=0
where, taking into account Proposition 6.5,
(N )
(n)
αi
=
BS
(x − c)N φN Pn , Qi
(N )
BS
=
(Qi , Qi )
hu, Pn F (N ) Qi i
.
k̃i
¿From the orthogonality of {Pn }n and since F (N ) Qi = 0 for i < N , we
(n)
deduce that αi = 0 when 0 ≤ i < r = max{N, n − N t}.
ii) Because of Proposition 6.7, the expansion of the polynomial F (N ) Qn
in terms of Pn is
F (N ) Qn (x) =
n+N
Xt
i=0
30
(n)
βi Pi (x).
(n)
The coefficients βi
fore
(n)
βi
=
can be computed using again Proposition 6.5, and there-
hu, Pi F (N ) Qn i
hu, Pi2 i
(N )
=
BS
(x − c)N φN Pi , Qn
.
ki
(n)
Finally, from the orthogonality of {Qn }n it follows βi
n − N (p + 1).
= 0 for 0 ≤ i <
¿From the symmetry of the linear operator F (N ) , we can obtain a difference–differential relation satisfied by the Sobolev orthogonal polynomials
with respect to the Sobolev bilinear form (2.1), where u is a semiclassical
linear functional.
Proposition 7.2 (Difference–Differential Relation) For every n ≥ N ,
the following relation holds
F (N ) Qn (x) =
n+N
Xt
(n)
γi Qi (x),
(7.3)
i=r
(N )
(n)
(n)
where r = max{N, n − N t}, γn+N t = F (n) and γr
=
BS
Qn , F (N ) Qr
k̃r
.
Proof. Consider the Fourier expansion of the polynomial F (N ) Qn in terms
of Qn which, by Proposition 6.7, is
F
(N )
Qn (x) =
n+N
Xt
(n)
γi Qi (x).
i=0
Then
(N )
(n)
γi
=
BS
F (N ) Qn , Qi
(N )
BS
(N )
=
BS
Qn , F (N ) Qi
(Qi , Qi )
k̃i
,
(n)
where we have used Theorem 6.6. Notice that γi = 0 for 0 ≤ i < N
(n)
and that the orthogonality of the polynomials {Qn }n leads to γi = 0 for
0 ≤ i < n − N t. So the result follows.
Remark. In formulas (7.1) and (7.3), when r = n − N t, the coefficients
(n)
(n)
αr and γr can explicitly be given by
αr(n) = F (r)
kn
,
k̃r
γr(n) = F (r)
k̃n
.
k̃r
Recall that the values of F (n) had been calculated in Theorem 6.11.
31
Acknowledgements
First and fourth authors wish to acknowledge the financial support by DGES
project PB96–0120–C03–02 and by the Universidad de La Rioja project
API-98/B12. Second and third authors wish to acknowledge the financial
support by Junta de Andalucı́a, G.I. FQM 0229, DGES PB 95–1205 and
INTAS–93–0219–ext.
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∗
Departamento de Matemáticas. Universidad de Zaragoza. 50009 Zaragoza
(Spain).
E–mail: [email protected], [email protected]
†
Departamento de Matemática Aplicada. Instituto Carlos I de Fı́sica
Teórica y Computacional. Universidad de Granada. 18071 Granada (Spain).
E–mail: [email protected], [email protected]
33